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61.
We establish a new estimate for the topological degree of continuous maps from the sphere into itself, which answers a question raised in Bourgain, Brezis, and Mironescu [Commun. Pure Appl. Math. 58 (2005) 529–551] and extends some of the results proved there, as well as in recent work by these authors (Lifting, degree, and distributional Jacobian revisited, http://ann.jussieu.fr/publications). To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005). 相似文献
62.
63.
Let A be a subset of a finite field
for some
prime q. If
for some > 0, then we prove the estimate
for some = () > 0. This is a finite field
analogue of a result of [ErS]. We then use this estimate to prove a
Szemerédi-Trotter type theorem in finite fields, and obtain a new estimate for
the Erdös distance problem in finite fields, as well as the three-dimensional
Kakeya problem in finite fields. 相似文献
64.
In this Note, we extend the results of Bourgain, Konyagin and Glibichuk to certain composite moduli q involving few ‘large’ primes. First a ‘sum-product’ theorem for subsets A of is obtained, ensuring that provided and A does not have a ‘large’ intersection with a translate of a subring. Next, exponential sum estimates are established. In particular nontrivial bounds are obtained for the exponential sums associated to a multiplicative subgroup , with applications to Heilbronn-type sums. To cite this article: J. Bourgain, M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 339 (2004). 相似文献
65.
Jean Bourgain 《Comptes Rendus Mathematique》2004,339(5):321-325
We establish a Mordell type exponential sum estimate (see Mordell [Q. J. Math. 3 (1932) 161–162]) for ‘sparse’ polynomials prime, under essentially optimal conditions on the exponents . The method is based on sum–product estimates in finite fields and their Cartesian products. We also obtain estimates on incomplete sums of the form for , under appropriate conditions on the . To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 339 (2004). 相似文献
66.
The main result is the following. Let be a bounded Lipschitz domain in , d?2. Then for every with ∫f=0, there exists a solution of the equation divu=f in , satisfying in addition u=0 on and the estimate where C depends only on . However one cannot choose u depending linearly on f. To cite this article: J. Bourgain, H. Brezis, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 973–976. 相似文献
67.
The study of sum and product problems in finite fields motivates the investigation of additive structures in multiplicative subgroups of such fields. A simple known fact is that any multiplicative subgroup of size at least q 3/4 in the finite field F q must contain an additive relation x + y = z. Our main result is that there are infinitely many examples of sum-free multiplicative subgroups of size Ω(p 1/3) in prime fields F p . More complicated additive relations are studied as well. One representative result is the fact that the elements of any multiplicative subgroup H of size at least q 3/4+o(1) of F q can be arranged in a cyclic permutation so that the sum of any pair of consecutive elements in the permutation belongs to H. The proofs combine combinatorial techniques based on the spectral properties of Cayley sum-graphs with tools from algebraic and analytic number theory. 相似文献
68.
Jean Bourgain 《Journal d'Analyse Mathématique》2014,124(1):297-307
It is shown that monotone Boolean functions on the Boolean cube capture the expected number of primes, under the usual identification by binary expansion. This answers a question posed by G. Kalai. 相似文献
69.
70.